二维三温热传导方程求解中的非线性迭代初值选取

由于二维三温热传导方程具有很强的非线性特性,因此采用全隐格式对该方程离散后,所得非线性代数方程组的求解将变得非常困难.针对二维三温热传导方程离散所得非线性代数方程组的迭代求解,提出了一种有效的选取初值的方法.对两种不同性质的介质进行数值实验,结果表明,所设计的初值选取方法不仅大大提高了计算效率,而且能够降低非线性解法器对时间步长的影响.     

分片光滑物态方程的能量方程非线性迭代解法

实际应用中的物态方程由分片光滑曲面拼接而成,拼接处存在间断.隐式求解相应的能量方程时,经常出现迭代收敛慢的情况和非物理解.本文通过构造对应的新的非线性问题,提出一种非线性迭代算法.该算法适用于求解有间断的分片光滑物态方程的非线性能量方程,其中引入一个度量能量变化的参数用于自动判断跳段是否发生,在求解时无需事先知道物态方程间断的位置,且能精确计算物态方程间断带来的能量盈亏,用于评估物态方程间断对能量的影响.典型算例验证了新算法具有稳定的收敛性,并给出符合物理规律的解.     

A new method based on the harmonic balance method for nonlinear oscillators

The harmonic balance (HB) method as an analytical approach is widely used for nonlinear oscillators, in which the initial conditions are generally simplified by setting velocity or displacement to be zero. Based on HB, we establish a new theory to address nonlinear conservative systems with arbitrary initial conditions, and deduce a set of over-determined algebraic equations. Since these deduced algebraic equations are not solved directly, a minimization problem is constructed instead and an iterative algorithm is employed to seek the minimization point. Taking Duffing and Duffing-harmonic equations as numerical examples, we find that these attained solutions are not only with high degree of accuracy, but also uniformly valid in the whole solution domain.     

一种新的重排方法在二维三温问题求解中的应用

针对二维三温问题离散化得到的稀疏线性方程组,提出一种新的重排技术(交替hyperplane重排),并结合ILU分解预条件技术,在Krylov子空间迭代法下进行测试.数值实验表明,在一定的填充模式及预处理消耗大致相同的前提下,使用交替hyperplane重排技术的迭代收敛效果明显优于红黑排序、hyperplane排序等方法.     

非线性波动方程的新数值迭代方法

提出了一种新的求解非线性波动方程的数值迭代法,它是一种半解析的方法.与完全的数值计算方法扰法相比,它能够考虑各阶谐波的相互作用,且能够满足能量守恒定律.用它研究了非线性声波在液体中的传播性质,结果表明,在微扰法适用的声强范围内迭代法也适用,在微扰法不适用的一个较宽的声强范围内迭代法依然适用.     

α能谱峰形函数及其拟合参数初始值获取

(1. 西南科技大学 核废物与环境安全国防重点学科实验室, 四川 绵阳 621010;2. 四川理工学院 化学与环境工程学院, 四川 自贡 643000;3. 成都理工大学 地质灾害防治与地质环境保护国家重点实验室, 成都 610059;4. 中国工程物理研究院 核物理与化学研究所, 四川 绵阳 621900) 根据能谱的形成机理,建立了一种能谱峰形函数,提出基于矩估计法确定该峰形函数的拟合参数初始值的方法。以数理统计为基础,分别通过峰形函数和离散谱峰数据计算均值、方差和三阶中心矩建立方程组求解获取拟合参数的初始值。通过连续气溶胶检测仪探测的氡子体能谱和低本底能谱仪探测的238Pu面源能谱对该峰形函数模型和初始值获取方法进行了测试。结果表明:从低重合度到高重合度,该峰形函数模型都能得到较好的应用;基于矩估计法确定该峰形函数的参数初始值能够使该峰形函数较好地拟合能谱数据。该方法在采用计算机自动能谱拟合分析中具有较强的实用性。     

Delayed models for simplified musical instruments

Most musical instruments contain, at their very basis, a continuous vibrating element (string or air column) which can be treated as a one-dimensional system. Its oscillation is obtained either through an initial condition or by means of a continuous energy input through a nonlinear device. In both cases and as a first approach, the excitation can be localized at one single point, and the continuous system can be considered as a linear one. The coupling between these two elements is often represented through a convolution integral. This convolution will be rewritten here in a way that different phenomena taking place in the continuous element (internal losses, radiation at the ends...) are separated. Different choices in the formulation of these processes and some mathematical manipulation will lead to either algebraic iterative or delayed differential equations. These equations are valid for any form of energy input. Once this energy input is defined, they can be used to simulate the behavior of different instruments in a more efficient way than that of traditional convolution. Moreover, these equations allow an analytical analysis of possible regimes using the tools of nonlinear dynamical systems (NLDS). The case of woodwinds will be emphasized throughout the paper, while that of strings will be presented briefly for the sake of completeness.     

利用空间光调制器实现计算全息三维显示

针对目前三维物体计算全息算法数据量大、计算速度慢及共轭像影响再现效果问题,提出一种全息体视图计算方法。根据人眼双目视差立体视觉原理,由摄像机获取三维物体不同角度的二维序列视差图像,通过计算全息算法得到多视角全息图,合成三维物体全息体视图。在迭代傅里叶变换算法的基础上,采用预设初始相位并增加反馈因子的方法,提高相息图的计算效率。基于液晶空间光调制器构建光学系统,对计算的全息体视图进行了光学再现。结果表明:该方法有效地排除了共轭像的干扰,相息图的迭代计算效率提高30%以上,再现图像与目标图像的结构相似度大于0.85。     

Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type

Lane–Emden equation is a nonlinear singular equation in the astrophysics that corresponds to the polytropic models. In this paper, a pseudospectral technique is proposed to solve the Lane–Emden type equations on a semi-infinite domain. The method is based on rational Legendre functions and Gauss–Radau integration. The method reduces solving the nonlinear ordinary differential equation to solve a system of nonlinear algebraic equations. The comparison of the results with the other numerical methods shows the efficiency and accuracy of this method.     

The residue harmonic balance for fractional order van der Pol like oscillators

When seeking a solution in series form, the number of terms needed to satisfy some preset requirements is unknown in the beginning. An iterative formulation is proposed so that when an approximation is available, the number of effective terms can be doubled in one iteration by solving a set of linear equations. This is a new extension of the Newton iteration in solving nonlinear algebraic equations to solving nonlinear differential equations by series. When Fourier series is employed, the method is called the residue harmonic balance. In this paper, the fractional order van der Pol oscillator with fractional restoring and damping forces is considered. The residue harmonic balance method is used for generating the higher-order approximations to the angular frequency and the period solutions of above mentioned fractional oscillator. The highly accurate solutions to angular frequency and limit cycle of the fractional order van der Pol equations are obtained analytically. The results that are obtained reveal that the proposed method is very effective for obtaining asymptotic solutions of autonomous nonlinear oscillation systems containing fractional derivatives. The influence of the fractional order on the geometry of the limit cycle is investigated for the first time.     

Étude numérique de l'influence du transfert radiatif sur la convection naturelle laminaire,bidimensionnelle, permanente,dans un espace annulaire d'axe horizontal,délimité par deux cylindres circulaires isothermes

Combined radiation and natural convection in a participating medium between concentric or vertically eccentric horizontal cylinders is investigated numerically. The annular medium is considered as a gray, emitting, absorbing, and isotropically scattering gas. The equations of steady, laminar, two-dimensional, thermal, natural convection are written by using a two-cylindrical coordinate system, the stream function, and the vorticity. The finite volumes method is used to discretize the coupled equations of momentum, energy, and radiative transfer. To solve the global nonlinear algebraic equations the successive-over-relaxation iterating scheme is applied. Numerical solutions are obtained for a Rayleigh number in the range 10³–10⁵ and radiation-conduction parameter ranging from 0 to ∞. The influences of radiation-conduction parameter, Rayleigh number and other parameters on flow and temperature distributions and heat transfer are discussed.     

二维三温能量方程组的高效代数解法

针对二维三温能量方程九点格式离散后形成的非线性方程组,研制了高效求解的代数解法器.主要思想是在部分Newton-Krylov(PNK)方法和Jacobi矩阵自由的Newton-Krylov(JFNK)方法的框架下,结合非精确Newton类方法和预条件Krylov子空间方法进行高效求解.数值结果显示,PNK方法比非线性块Gauss-Seidel方法快6倍以上,在PNK框架下比较了3种预条件子和4种Krylov子空间方法,得出不同组合的最佳方案.还比较了JFNK方法和PNK方法.     

Mapping deformation method and its application to nonlinear equations

An extended mapping deformation method is proposed for finding new exact travelling wave solutions of nonlinear partial differential equations (PDEs). The key idea of this method is to take full advantage of the simple algebraic mapping relation between the solutions of the PDEs and those of the cubic nonlinear Klein－Gordon equation. This is applied to solve a system of variant Boussinesq equations. As a result, many explicit and exact solutions are obtained, including solitary wave solutions, periodic wave solutions, Jacobian elliptic function solutions and other exact solutions.     

Nonlinear coupled-mode equations

Nonlinear coupled-mode equations governing the modal coupling of a two-mode coupled system (such as twin core couplers) are integrable; power swapping in such a system follows a periodical manner and can be expressed analytically. When three or more modes (for systems such as multiple-core couplers) are involved, the nonlinear coupled-mode equations are no longer integrable and chaotic power swapping is expected. A numerical approach is required, in general, to solve such nonlinear coupled systems involving the coupling of three or more modes. We find, however, that for certain structural configurations, such as triple-core couplers with the cores arranged in the shape of an isosceles triangle, the nonlinear coupled-mode equations for multiple-core couplers can be solved analytically under a resonant condition. The analytical solution indicates that power swapping among, for example, the three cores placed in the shape of an isosceles triangle can be aperiodic at high power, although power may flow from core to core periodically at low power.     

A Novel Method to Solve Nonlinear Klein-Gordon Equation Arising in Quantum Field Theory Based on Bessel Functions and Jacobian Free Newton-Krylov Sub-Space Methods

The Klein-Gordon equation arises in many scientific areas of quantum mechanics and quantum field theory.In this paper a novel method based on spectral method and Jacobian free Newton method composed by generalized minimum residual(JFNGMRes) method with adaptive preconditioner will be introduced to solve nonlinear Klein-Gordon equation. In this work the nonlinear Klein-Gordon equation has been converted to a nonlinear system of algebraic equations using collocation method based on Bessel functions without any linearization, discretization and getting help of any other methods. Finally, by using JFNGMRes, solution of the nonlinear algebraic system will be achieved. To illustrate the reliability and efficiency of the proposed method, we solve some examples of the Klein-Gordon equation and compare our results with other methods.     

分子高激发振动的经典代数方法研究——以C_2H_2的C-H弯曲振动为例

本文以目前探讨得较多的C2H2的CH弯曲振动为例,说明如何应用代数方法来研究分子的高激发振动。由于分子的高激发振动态具有很强的模间非线性偶合以及能量的传递,传统的动力学方法似乎很难有效地用来研究其性质。问题的核心是高激发振动态由于其量子数很大,因此具有经典(或半经典)的性质。同时模间能量的传递可以用二次量子化算子来表示,而这些算子所具有的代数性质,使得人们可以用几何的概念来描述其性质。因此,整个问题就变为用几何的观点来分析分子的高激发振动态。最后,我们用所得的经典的代数哈密顿量和哈密顿方程对CHtrans弯曲和cis弯曲振动模间能量的传递速度与体系所含能量之高低的关系做了探讨     

A method for nonlinear modal analysis and synthesis: Application to harmonically forced and self-excited mechanical systems

The recently developed generalized Fourier–Galerkin method is complemented by a numerical continuation with respect to the kinetic energy, which extends the framework to the investigation of modal interactions resulting in folds of the nonlinear modes. In order to enhance the practicability regarding the investigation of complex large-scale systems, it is proposed to provide analytical gradients and exploit sparsity of the nonlinear part of the governing algebraic equations.     

用代数多重网格法求解一维分裂格式的Euler方程

提出了代数多重网格法(AMG)的一种新算法。新算法改进了插值公式和粗网格方程,并把它应用到求解一维的分裂格式Euler方程。数值结果表明,对于具有高CFL条件数的Euler方程,代数多重网格法可以求解;对于Gaus-Seidel方法求解不能收敛的代数方程组,代数多重网格法求解可以收敛。新算法改进了代数多重网格法的收敛性和扩展了它的应用范围,数值结果表明了它的有效性和强壮性。     

A framework for iterative analysis of non-classically damped dynamical systems

In this paper, we propose a general iterative framework to solve the dynamic problem for linear systems with non-classical viscous damping. A systematic approach is used to derive families of stationary iterative schemes that, as an instance of particular interest, decouple the equations of motion for numerical study of the system response. For such schemes, we present a detailed convergence analysis and propose several solution strategies suitable for a broad class of systems. These techniques are based on spectral analysis of particular iteration matrices arising in the derivation and aim at optimizing the convergence performance of the method. We demonstrate that the proposed systematic framework, based on a novel application of the homotopy analysis method, generalizes iterative schemes previously reported in the literature and, importantly, provides a unified perspective for the study of iterative solutions of dynamic problems. Further, we establish a connection between our results and the theory of iterative schemes for algebraic linear systems, thus providing insights on convergence results and applicability of the method. Numerical examples illustrate the effectiveness of the approach and indicate future research directions.